Osamurtokehitelmää

Osamurtokehitelmää, often translated as partial fraction decomposition, is a technique used in calculus and algebra to simplify complex rational functions into a sum of simpler fractions. A rational function is a ratio of two polynomials, P(x)/Q(x). Partial fraction decomposition is applicable when the degree of the numerator polynomial P(x) is less than the degree of the denominator polynomial Q(x).

The process involves factoring the denominator Q(x) into its irreducible factors. These factors can be linear (of the form ax + b) or quadratic (of the form ax^2 + bx + c that cannot be factored further into real linear factors). For each factor in the denominator, a corresponding term is created in the partial fraction decomposition. Linear factors of the form (ax + b)^n lead to terms of the form A/(ax + b) + B/(ax + b)^2 + ... + K/(ax + b)^n, where A, B, ..., K are constants to be determined. Irreducible quadratic factors of the form (ax^2 + bx + c)^m lead to terms of the form (Ax + B)/(ax^2 + bx + c) + (Cx + D)/(ax^2 + bx + c)^2 + ... + (Px + Q)/(ax^2 + bx + c)^m.

Once the general form of the decomposition is established, the unknown constants are found by equating the original rational function to the sum of the partial fractions and clearing the denominators. This typically results in a system of linear equations that can be solved for the constants. The primary application of partial fraction decomposition is in integration, as the integrals of the simpler fractions are often much easier to evaluate than the integral of the original complex rational function. It can also be useful in solving differential equations and in analyzing electrical circuits.